Pressure & Fluid Statics

Pressure:

Pressure is defined as: ' Normal force per unit area of body'.

There are different pressures which we described ahead:

• Atmospheric Pressure  ⟺  Pressure exerted by atmosphere.
• Absolute Pressure  ⟺  The actual pressure at a given point. It is calculated with respect to absolute zero pressure.
• Gage Pressure  ⟺  It is difference between the absolute pressure and the local atmospheric pressure.

• Vacuum Pressure  ⟺  Pressure below atmospheric pressure. 

Pressure At A Point In A Fluid:

For the pressure at a point in fluid, consider a triangular area of fluid. Consider a free body diagram with in a fluid mass.

The force and weight components along Z-axis is given by:

So, pressure at a point of a fluid at rest or motion is independent of direction as long as there are no shearing stresses present.

Types Of Forces:

There are two types of forces which are described ahead:

1. Body Forces  ⟺  Forces which act throughout the entire body and are proportional to the volume of the body.
2. Surface Forces  ⟺  Forces which act only on surfaces and are proportional to the surface area of the body.

Basic Equation For Pressure Field:

Consider a small rectangular element of incompressible and static fluid removed from any position within the mass of the fluid. 

Using Taylor Series for the pressure at different points:

Since, 'h' is very small so neglecting its 2nd and higher powers.

Now, consider Body forces:

A. For Static(Acceleration = 0):

Consider the General Equation of Motion of the fluid, we get:

So, the pressure is only dependent on vertical axis that is Z-axis in this case.

1. For Homogeneous Incompressible Fluid:

Consider the final result of static case of pressure field, we get:

2. For Compressible Fluid:

Consider the compressibility of fluid. Applying the final result of static case of pressure field, we get:

a. For Stratosphere(Isothermal region of Atmosphere):

Now from Ideal gas equation:

b. For Toposhere(Linear Temperature Variation Region):

Measurement Of Pressure:

There are various measuring instrument of pressure, namely:

1. Barometer & Manometer
2. U-Tube Manometer 
3. Piezometer Tube 
4. Inclined Tube Manometer

Mechanical & Electronic Pressure Measuring Devices:

  There are numerous pressure measuring devices, namely:

1. Aneroid Barometer  ⟶  To find the atmospheric pressure.
2. Pressure Transducer  ⟶  Calculate pressure with the help of electric signals and LVDT (Linear Variable Differential Transformer + Bourdan Tube Gage) for better efficiency. 
3. Bourdan Pressure Gage  ⟶  To calculate local atmospheric pressure.

Hydrostatic Forces On A Plane Surface:

When a body is submerged into a fluid, force is exerted on the surface of body is called Hydrostatic Forces.

Consider a rectangular bar submerged into liquid having area 'A'. The hydrostatic forces applied on the body is given by:

 Location Of Center Of Pressure:

First considering the scenario about X-axis. Moment of resultant force must be equal to the moment of the distributed pressure force:

and,

• For Uniform distribution of pressure  ⟶  Center of pressure = Centroid.
• For Non-Uniform distribution of pressure  ⟶  Center of pressure is always lower than centroid.

Pressure Prism:

 The resultant force must lie on on the centroid of the pressure prism, which is located along the vertical axis of symmetry of the surface.

Hydrostatic Forces On A Curved Surface:

Forces will be concentric or radial. Direction and magnitude of forces will change from point to point.

Now considering the forces and finding the horizontal and vertical forces on a curved surface, we get:

B. For Motion:

Now the General Equation of pressure field is given by:

Now, the separable form of the above equation is given by:

a. Free-Fall (Without Friction):

When free-fall, acceleration along X and y axis are zero and acceleration along Z-axis is -g.

b. Lifting Upward:

When free-fall, acceleration along X and y axis are zero and acceleration along Z-axis is +g.

c. Moving Along An Axis With Constant Acceleration:

Since, no motion of fluid along Y-axis, so acceleration along Y-axis is zero.

d. Rigid Body Rotation:

• If you want to work on angle whether in 2D or 3D, it is easy to use Polar Coordinate System (r, θ, z).
• Since, cylindrical container is symmetrical about Z-axis, so no dependence on 'θ'. So change in pressure with respect to transverse axis is zero.

The total pressure is given by:

Buoyancy, Flotation & Stability:

According to the Archimedes Principle, An upward buoyant force that is exerted on a body immersed in a fluid, whether partially of fully submerged, is equal to the weight of the fluid that the body displaces and acts in the upward direction at the center of mass of the displaced fluid.

During the position of floating, there are two forces applied on the body:

1. Weight acting on its center of gravity.
2. Upthrust force acting upward on its center of buoyancy.

The difference between these forces is called Buoyant force.

• Centroid of Displaced Volume is called Center of Buoyancy.
• If Upthrust < Weight, then body sinks.
• If Upthrust > Weight, then body floats.

Principle Of Flotation:

Principle of flotation means at what circumstances a body will float. There are three condition on which a body's floating behavior is described:

• If weight of body = upthrust due to liquid  ⟷ body is partially submerged but floats.
• If weight of body > upthrust due to liquid ⟷ body sinks.
• If weight of body < upthrust due to liquid ⟷ body will float on the liquid.

For Floating bodies, the weight of the body is equal to the buoyant force which is the weight of the displaced liquid. Therefore:

From above equation, one can know that the body which will immersed in liquid will float, sink or submerged. So:

1. If density of both are same, then body will submerged.
2. If fluid density > body density, then body will float.
3. If fluid density < body density, then body will sink.

Stability:

It characterizes response of an object to small disturbance like air or waves in ocean. It is the most important application of buoyancy which deals with the body immersed in liquid will float, submerged or sink. 

There are three cases to describe the behavior or state of body:

1. Stable Object  ⟹  Which restores its original position.
2. Unstable Object  ⟹  Which gains new position after having small disturbance.
3. Neutral Object  ⟹  Which remains in disturbed position after small disturbance.

Stability Of Submerged Body:

The stability of submerged or floating body depends on the relative position of center of buoyancy and center of gravity.

• If Buoyancy Center > Gravity Center  ⟺  Object is stable, body will float.
• If Buoyancy Center < Gravity Center  ⟺  Object is unstable, body will sink.
• If Buoyancy Center = Gravity Center  ⟺  We need define a new word which tells us whether the body is stable or unstable which is Meta-Center.

Metacenter:

 The point which is obtained by joining the symmetrical line drawn from the center of gravity and vertical line drawn from center of buoyancy.
  

 By knowing the metacenter and comparing with center of gravity, we need to define the stability of object.

• If Metacenter(above) > Center of Gravity  ⟺  distance is positive and object is stable.
• If Metacenter(below) < Center of Gravity  ⟺  distance is negative and object is unstable. 

References:

• Materials From Class Lectures + Own Knowledge + Book named Fundamentals of Fluid Mechanics by Munson, Young and Okiishi's (8th Edition).
• Photos from Google Images.
• Videos from YouTube + Google.